Plasmonic Nano-Oven by Concatenation of ... - ACS Publications

Plasmonic Nano-Oven by Concatenation of Multishell Photothermal Enhancement Lijun Meng,†,‡ Renwen Yu,† Min Qiu,‡ and F. Javier García de Abajo*,†,§ †

ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310027, China § ICREA-Institució Catalana de Recerca i Estudis Avançats, Passeig Lluís Companys 23, 08010 Barcelona, Spain ‡

S Supporting Information *

ABSTRACT: Metallodielectric multishell nanoparticles are capable of hosting collective plasmon oscillations distributed among different metallic layers, which result in large near-field enhancement at specific regions of the structure, where light absorption is maximized. We exploit this capability of multishell nanoparticles, combined with thermal boundary resistances and spatial tailoring of the optical near fields, to design plasmonic nano-ovens capable of achieving high temperatures at the core region using moderate illumination intensities. We find a large optical intensity enhancement of ∼104 over a relatively broad core region with a simple design consisting of three metal layers. This provides an unusual thermal environment, which together with the high pressures of ∼105 atm produced by concatenated curved layers holds great potential for exploring physical and chemical processes under extreme optical/thermal/pressure conditions in confined nanoscale spaces, while the outer surface of the nano-oven is close to ambient conditions. KEYWORDS: metallodielectric nanoplasmonics, thermoplasmonics, multishell nanoparticles, near-field enhancement, optical heating Metallodielectric core−shell nanoparticles 36 produced through colloid synthesis have been widely investigated for applications such as nanolasing,37 photothermal therapy,27 solar vapor generation,38,39 and near-field concentration.35 Beyond conventional core−shell structures, multishell nanoparticles consisting of several alternating layers of metal and dielectric materials have been successfully synthesized,34,40−42 and their optical properties well understood as the result of hybridization of plasmons residing at the different metal components.34,35,40 Careful engineering of the resulting nanomatryoshka structures has revealed novel phenomena such as superscattering43 and plasmon Fano resonances.44 However, a thorough study of multishell nanoparticles with more than two metallic shells, and in particular the thermoplasmonic properties offered by such structures, is still missing. In this work, we combine optical Mie theory and the twotemperature model, including the influence of the temperaturedependent thermal boundary conductance (TBC, also known as Kapitza conductance45−48), to efficiently design a plasmonic nano-oven, in which we optimize both optical and thermal responses. Specifically, we show that nanoscale thermal

T

he interaction between metal nanostructures and resonant light has been widely explored during the past decades to achieve a remarkable degree of control over the optical intensity at deep-subwavelength scales.1 This interaction is mediated by collective oscillations of conduction electrons known as surface plasmon resonances, which are characterized by a pileup of induced charges at the metal surface accompanied by huge local field enhancements. A large deal of work has been devoted to investigate the ability of plasmons to boost different processes of interest (e.g., photochemistry,2−4 high-harmonic generation,5,6 perfect light absorption,7−9 and fluorescence emission10,11) and improve applications to sensing and light management (e.g., SERS-based biosensing,12−15 light modulation,16,17 structural coloring,18,19 and photodetection20−22). The plasmon energy is partially dissipated as Joule heat, which for strong field localization can lead to considerable concentration of temperature increase. This effect constitutes the basis of the emerging field of thermoplasmonics.23,24 It has also been exploited for optically assisted drug delivery,25,26 cancer therapy,27−29 optical switching,30,31 and heat-assisted photochemistry.32,33 In all of these applications, optimum designs of plasmonic structures are crucial to produce extraordinary field enhancement at designated spatial locations. A particularly interesting class of such structures is provided by metallic shells.27,34,35 © 2017 American Chemical Society

Received: April 7, 2017 Accepted: July 20, 2017 Published: July 20, 2017 7915

DOI: 10.1021/acsnano.7b02426 ACS Nano 2017, 11, 7915−7924

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dipolar modes of several of the gold shells in a constructive manner, so that inner layers are exposed to stronger optical fields and the core sees a large light intensity. This type of geometry has been previously explored for near-field concentration in multishells.35 This is a cascading effect similar to the one previously studied for self-similar particle chains.49 The increase in field intensity is also accompanied by higher absorption due to Joule losses, therefore resulting in an increase of temperature at the core of the structure, which is the main task of the nano-oven. This heat is eventually dissipated through the thermal conduction of the layers, until it is eventually released to the surrounding water. We remark that there is a beneficial effect arising from the presence of multiple metal/dielectric interfaces: each of them presents some resistance to the passage of heat, thus resulting in an efficient barrier that prevents heat at the core from rapidly escaping from the structure. In order to simulate the optical and thermal responses of the nano-oven, we construct semianalytical models, as discussed in detail in the Methods section. The optical response is expressed as a generalization of Mie theory51,52 adapted to multishell spheres. This approach is fast and accurate compared with alternative numerical methods, thus allowing us to explore a large number of geometrical configurations. The thermal response is formulated as a set of analytical nonlinear equations (see Methods) that describe heat exchange processes in a layer by layer fashion. We consider a monochromatic cw external illumination, so that the structure operates under steady-state conditions. We adopt the twotemperature model and define electron and lattice temperatures at each of the metallic layers. For simplicity, and because the thermal conductivity is large in gold compared with the surrounding silica and water media, we assume these two temperatures to be uniform within each metal layer. The temperatures are the unknowns of our set of nonlinear equations, together with the dielectric and water temperatures right at their interfaces with the metal. Heat flow within the nano-oven is mediated by several key processes that are sketched in the inset of Figure 1a. First of all, light is absorbed at the metallic layers, resulting in a positiondependent heat power density given by (in Gaussian units)

management can be achieved by simultaneously engineering multiple thermal boundary barriers and the plasmonic near-field enhancement. Our nano-oven design features three appealing properties: (i) it efficiently concentrates the electric field into the core region as a result of electrostatic cascading,35,49 reaching a relative intensity enhancement of >104; (ii) the optical field enhancement at the core region, accompanied by multiple thermal barriers originating in the TBC at the interfaces between metal and dielectric,50 render high core temperatures with relatively moderate illumination intensities; (iii) the accumulation of surface tensions and stresses at multiple interfaces results in an elevated pressure at the core region (∼105 atm). The nano-oven thus provides an excellent platform for the exploration of chemical and physical phenomena under the extreme optical, thermal, and pressure conditions that are available at its core using attainable continuous-wave (cw) illumination conditions.

RESULTS AND DISCUSSION Figure 1a presents a sketch of a metallodielectric multishell nanoparticle that acts as a nano-oven. It is formed by alternating gold and silica shells, embedded in a water environment. We aim at simultaneously exciting plasmon

pabs (r) =

ω Im{ϵ(r, ω)}|E(r)|2 2π

(1)

where ϵ(r,ω) is the position- and light-frequency-dependent permittivity in the local-response approximation, while E(r) is the amplitude of the optical electric field (i.e., E(r)e−iωt + E*(r)eiωt gives the full time-dependence of the electric field); eq 1 reveals that only the gold can directly couple optical energy into heat through its nonvanishing Im{ϵ}, dominated by electronic excitations. Coupling between electronic and lattice (phonons) degrees of freedom at each gold layer is described through a coefficient gel.53 Finally, the noted thermal barriers at each metal/dielectric interface are described through the TBCs Gel and Gll associated with two different channels, involving the coupling of electronic (Gel) and lattice (Gll) heat in the metal to lattice heat in the dielectric. All of these parameters are incorporated in our model, as explained in detail in the Methods section. Before discussing the multishell nano-oven, it is instructive to examine the steady-state thermal performance of a homogeneous gold sphere in water (Figure 2). We use a similar analysis as in the Methods, which extends the results of previous work24

Figure 1. Extraordinary optical, thermal, and pressure conditions in a plasmonic nano-oven. (a) Schematic view of a multishell nanoparticle made of alternating metallic and dielectric layers. The inset shows different heat transfer channels: coupling between electrons (e) and lattice modes (l) inside a metal layer (mediated by the volumetric transfer coefficient gel) and coupling between these degrees of freedom and lattice modes in the surrounding dielectrics through thermal boundary conductances (TBCs) Gel and Gll (see Methods). (b−d) We illustrate the large optical field enhancement (b), efficient temperature increase (c), and highpressure increase (d) in a plasmonic nano-oven composed of N = 3 gold layers intercalated with silica and immersed in water under ambient conditions. The incident light has an intensity of 1 GW/m2 and a wavelength of 690 nm. 7916


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Figure 2. Thermal response of small metal nanoparticles. (a) Radial distribution of the temperature for a gold nanosphere in water (5 nm radius) normalized to the optically absorbed power. The environment temperature is T0 = 300 K. (b) Under the same conditions as in (a), dependence of various temperatures on particle radius: electron and lattice gold temperatures Te and Tl, and water temperature right outside the particle, Tw.

Figure 3. Optical and thermal properties of optimized nanoparticles with single (a), double (b), and triple (c) gold shells. In each row, we show from left to right a scheme of the nanoparticle, normalized extinction and absorption cross-section spectra, the electric near-field intensity distribution under plane-wave illumination at a wavelength of 690 nm, and the absorption power density pabs generated at the same wavelength with a pump intensity of 1 GW/m2. The power color scale is saturated at 1.6 nW/nm3 for clarity.

in order to incorporate thermal barriers.54−56 Three combined effects contribute to create a high temperature in both the lattice (Tl) and electron (Te) subsystems of the gold: (i) the small surface area limits the rate of heat evacuation, thus resulting in a jump of temperature Tw in the water right outside the particle relative to ambient temperature T0 given by24

Tl − Tw =

P abs 4πR2Ggold/water

(iii) finally, the electron−lattice coupling, described through the coefficient53 gelgold= 3 × 1016 W m−3 K−1, produces an additional jump

P abs T − T0 = 4πRκ water

Te − Tl =

w

3P abs el 4πR3ggold

Notice that under steady-state conditions the absorbed power is conserved in these expressions during its flow from the light to the electrons, then to the lattice, and finally to the water. The temperature jump can be significant even for a gold nanoparticle of 5 nm radius (Figure 2a), while it is boosted as the radius becomes smaller (Figure 2b). This is an effect that deserves further investigation from the experimental viewpoint.

which scales linearly with the total absorbed power Pabs and is inversely proportional to the particle radius R and the thermal water conductivity κwater = 0.6 W m−1 K−1; (ii) additionally, the thermal barrier at the interface produces a jump in the gold lattice temperature Tl, expressed in terms of the gold/water TBC57 Ggold/water = 105 MW m−2 K−1 as 7917


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Figure 4. Temperature profiles of optimized nanospheres with single (a, d), double (b, e), and triple (c, f) metal shells under different external pump intensities. We show results for multishells made of gold (a−c) and aluminum (d−f), illuminated at a resonant light wavelegth of 690 and 300 nm, respectively. The electron (dashed curves) and phonon (solid curves) temperatures are assumed to be uniform within each metal shell (shadowed regions).

the lack of direct connection between optimum heating and maximum field enhancement, we observe that the inner gold layer is the hottest one in Figure 1, but this is not where the field enhancement reaches its maximum. Additionally, the accumulation of pressure at each of the interfaces leads to a value of ∼7 GPa at the core (see Figure 1d and Methods), thus contributing to create extreme conditions of optical field, temperature, and pressure near the core of the nano-oven. Further insight into the effect of multiple layers can be gained by examining the behavior of systems with an increasing number of shells. We consider particles with N = 1−3 gold shells in Figure 3, optimized to operate at a fixed light wavelength of 690 nm. The radial distances at the gold/silica interfaces correspond to the numerical labels indicated in the horizontal axes of Figure 4a−c. The normalized extinction and absorption cross sections (curves in Figure 3) show that all three nanostructures present optical resonances around the targeted wavelength (vertical dashed line). Inspection of the electric near-field distributions reveals that these are dipolar modes. Due to the cascading effect mentioned above, the maximum field enhancement increases with the number of gold shells, reaching a maximum >104 for N = 3. Incidentally, similar results are obtained by replacing gold with silver, with roughly a 10-fold increase in maximum intensity (see Figure S2 in the SI). From the near-field intensity distributions, we readily calculate the generated heat power density using eq 1. The results are shown in the rightmost plots of Figure 3 for a pump intensity of 1 GW/m2. The maximum of pabs is ∼1−7−22 nW/ nm3 for the N = 1−2−3 structures, demonstrating the expected cascading increase in heat power density with increasing N. This high heat power density, supplemented by the thermal barriers of the alternate metal/dielectric interfaces, contributes to produce a large temperature increase at the core. We stress again that the structure optimized to render maximum field enhancement is not necessarily optimized in terms of temperature increase (see Figure S1, SI): smaller nanoparticles typically produce larger field enhancement, and therefore, also larger heat power-density generation; now, enlarging the size of

The temperature jumps in Figure 2 are normalized to the absorbed power, which for a sufficiently small sphere is given by the electrostatic expression P abs = 24π 2(R3/λ)ϵ3/2 water I0 Im{ − 1/(ϵgold + 2ϵwater )}

in terms of the light intensity I0 and the permittivities of gold and water (e.g., Pabs ≈ 1 μW for R = 5 nm, I0 = 12 GW/m2, and a resonant light wavelength λ = 522 nm). The same three mechanisms are at work in the nano-oven, operating in a constructive concatenation through the multiple shells, and supplemented by the noted optical cascading effect, which contributes to accumulate more light absorption near the nanooven core, as we show next. We now return to the nano-oven structure and use the thicknesses of the different gold and silica layers as geometrical parameters that allow us to optimize its performance. Focusing on a nano-oven consisting of three gold layers intercalated with silica, after a thorough examination of these parameters, with the additional constraint that metal layers are at least 3 nm in thickness, we obtain the optimized structure considered in Figure 1b−d (see radii below), operating at a light wavelength of 690 nm. Figure 1b illustrates that this structure is capable of concentrating the optical field near the core region, where the field-intensity enhancement exceeds 104. This is remarkable considering that light is impinging from outside the structure, so that it has to cross several metal layers to reach the core region. Incidentally, the intensity profile suggests a dominant effect of dipolar modes. As expected, this results in a substantial increase in temperature (Figure 1c), despite the fact that we are considering a moderate pump intensity of 1 GW/m2. We note that this is an optimum structure from the thermal viewpoint, but it is not necessarily optimum from the optical viewpoint. Indeed, as illustrated in Figure S1 of the Supporting Information (SI) for a simple core−shell structure, the geometrical parameters needed to optimize optical heating are not the same as those required to produce the largest field enhancement in the core region. As a further manifestation of 7918


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ACS Nano the structure causes a reduction in field enhancement, which is however compensated by the increase in particle volume, leading in some cases to higher total heat power (see Figure S3, SI). In Figure 4a−c, we present calculated temperature profiles for the same three gold/silica nanoparticles as in Figure 3. Under constant pump power, the maximum temperature increases with the number of metal layers, leading to ∼600 °C at the core of the N = 3 structure for a moderate pump intensity of 2 GW/m2 (Figure 4c, red curve). Additionally, we observe a clear difference (∼80 °C) between electron and lattice temperatures in the innermost gold shell. Recently, aluminum has attracted much attention as a plasmonic material58−60 because it is abundant, low-cost, and CMOS-compatible, and because it operates at shorter wavelengths that can reach the UV regime, which can be important for photocatalytic processes.3 We examine in Figure 4d−f the performance of nano-ovens formed by aluminum/silica structures, now optimized to operate at 300 nm light wavelength (see also Figure S4, SI). The obtained temperature increase is lower than for gold under constant pump power, an effect that we attribute to both the weaker capability of aluminum to confine the electric field and the much larger TBC of the latice−lattice channel at aluminum/silica interfaces (∼7fold increase with respect to gold; see details in Methods). As shown in Figure 4d−f, the high efficiency of heat conduction across the aluminum/silica interfaces substantially reduces the temperature jumps, and therefore, the temperature increase at the core mainly originates from the poor thermal conductance of the silica shells. Although the central temperature of the aluminum N = 3 nano-oven reaches 200 °C under a pump power of 2 GW/m2, which is only one-third of that in the gold nano-oven, it is still much higher than that in the aluminum core−shell structure, thus demonstrating its advantage to enhance the photothermal effect in the UV regime.

We have also considered aluminum-based nano-ovens because they can operate in the UV regime, although their performance is poorer than gold. We expect that silver should perform even better than gold because of its lower optical absorption (i.e., the field enhancement should be larger for constant illumination intensity), although its melting point is also lower, thus presenting potential problems when elevated temperatures are considered. As an interesting avenue, the use of refractory plasmonic materials with higher melting point could push the level of temperature increase even further.64 Beyond this work, we expect that there is plenty of room to further improve the performance of nano-ovens in terms of thermal management. For example, by making rough interfaces65 or by inserting organic molecules between otherwise closely attached metal/dielectric layers,66 one can significantly reduce the TBC, thus leading to higher temperature increases under constant light pumping intensity. A similar effect could be exploited in the outermost metal/water interface, for example by decorating it with hydrophobic groups.67,68 Additionally, the TBCs of interfaces formed by plasmonic metals and other dielectric materials beyond silica are only poorly known, thus demanding further exploration of their performance as nano-oven dielectrics, which could contribute to elucidate their thermal properties with application in the optimization of heat management in nanoscale devices. Overall, the extreme conditions created at the core of nanoovens provide a fantastic platform for the study of physical and chemical processes in which cooperative optical-field-, temperature-, and pressure-driven effects can take place.69−72 For example, band gaps and fluorescence intensities of organic and inorganic crystals can be tuned by applying different levels of pressure.69−71 Also some phase transitions occur under extreme pressure and temperature conditions.72,73 These effects could have potential application to optical memories, as already explored for homogeneous gallium nanoparticles.74 The study of these types of transitions taking place at the core of nanoovens could be facilitated by the availability of colloid methods for the synthesis of the multishells under consideration.34 As a related direction for exploration, phase transitions in the multishells themselves should result in even more dramatic changes in pressure. Similar to what has been recently demonstrated for larger multiparticle structures,75,76 a single multishell could eventually explode like a nanobomb, ignited for example by an intense pump laser pulse, which could be used to kill cancer cells.75,76 In a different direction, nano-ovens could be used to explore chemistry in confined nanoscale spaces, with conditions recreating those of stellar chemistry, while the outer surface of the structure is close to ambient conditions. Likewise, nanoovens provide a fantastic in situ platform for the exploration of hazardous reactions, which could take place inside the core region (e.g., by filling it with the appropriate reactants) while the outer layers prevent the involved substances from leaking out, with external light being used to control the temperature and probe the state of the reactions. As a specific example, the polymerization of cyanogen, which is a rather toxic substance in its standard phase, requires high pressure (∼GPa) and temperature (∼400 °C), which are typically achieved in a diamond anvil cell (DAC).77,78 When using the nano-oven, the reaction of cyanogen polymerization could take place inside the core region, which is isolated from the environment. Importantly, this reaction could be triggered using a moderate input light power, thus resulting in higher energy efficiency

CONCLUSIONS In conclusion, we have designed an optimized plasmonic nanooven based upon gold/silica multishells driven by visible light. Compared with a conventional core−shell nanoparticle, our proposed nano-oven is capable of simultaneously producing a huge electric field enhancement, a large temperature increase under moderate pump intensities, and a large pressure in the core region. We anticipate that comparison of model calculations and future experiments could shed light on the different thermal parameters involved in the nano-oven, including the effect of nanostructuration and the role played by the detailed phonon dispersion relations.61 Additionally, we find our simulations to be rather robust with respect to the assumption of the twotemperature model and the detailed temperature dependence of the TBCs. Indeed, as we show in Figure S5 of the SI, similar results are obtained by using a one-temperature model and temperature-independent TBCs. In this respect, there is a dispersion of reported values for the measured TBCs depending on sample preparation conditions62 and interfacial imperfections.47 We have used a rather high value of the TBCs, which leads to a conservative underestimate of the temperature increase. In fact, good agreement with optically induced melting of nanoparticles has been described using higher temperatureindependent TBC values,63 although again the results might depend on synthesis conditions. 7919


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Figure 5. (a) Mie scattering coefficients at a spherical interface. (b) Parameters used for the modeling of the thermal response of multishells. We show one of the metal layers (j), along with its corresponding radii, temperatures, and TBCs. (c, d) Temperature-dependence of the TBCs in gold−silica (c) and aluminum−silica (d) interfaces. The temperature is taken as that of the metal conduction electrons. n ,o n , n + 1 n ,i n + 1,o [Elm Elmν]θ(a − r ) + tlnν, n + 1Elm ν + rlν ν θ(r − a),

than with DAC-based methods. As a way of inserting the reactants inside the core region, we suggest the fabrication of hollow mesoporous metallic nanoshells.39 Finally, to collect the products, one could break the nano-oven by applying an intense laser pulse, similar to the nanobomb concept that we discuss above. This could be combined with ultrafast optical pumping in order to produce much larger electron temperatures, reaching several thousand degrees without material damage because the electronic heat capacity is much smaller than the lattice one. The dynamics of heat transfer from the electrons and the lattice, which has been studied for homogeneous media particles,79,80 will then be strongly affected by thermal barriers in multishell nano-ovens, and this could be eventually interesting for hot-electron production and their use in plasmon-driven photochemistry.81

n ,i n + 1,i n + 1, n n + 1,o tlnν̃ + 1, n Elm Elmν ]θ(r − a) νθ(a − r ) + [Elmν + rlν̃

where θ is the step function. Notice that l, m, and ν are conserved upon reflection or transmission at the interface. Additionally, the scattering coefficients are independent of m. They are obtained from the continuity of the parallel electric and magnetic field components at the interface. Their detailed expressions for the r = a interface separating generic media 1 (within r < a) and 2 (within r > a) are as follows: (+) (+) (+) (+) rl12 M = A {ρ2 hl (ρ1)[hl (ρ2 )]′ − ρ1hl (ρ2 )[hl (ρ1)]′} (+) (+) tl12 M = Aρ1{hl (ρ1)jl′(ρ1) − jl (ρ1)[hl (ρ1)]′} (+) (+) (+) (+) rl12 E = B{ϵ1hl (ρ1)[ρ2 hl (ρ2 )]′ − ϵ 2hl (ρ2 )[ρ1hl (ρ1)]′} (+) (+) tl12 E = Bϵ1ρ1{hl (ρ1)jl′(ρ1) − jl (ρ1)[hl (ρ1)]′}

METHODS Optical Simulations. We extend the Mie theory51,52 to rigorously simulate the optical response of multishell nanoparticles. This analytical method allows us to carry out fast calculations in order to search for optimized structures. More precisely, we focus on monochromatic light of frequency ω and consider the spherical waves hosted by each homogeneous spherical layer n in the multishell. We have outgoing spherical waves,82,83

rl21 ̃ M = A{ρ2 jl (ρ1)jl′(ρ2 ) − ρ1jl (ρ2 )jl′(ρ1)} ̃ M = Aρ2 {hl(+)(ρ2 )j′(ρ2 ) − j (ρ2 )[hl(+)(ρ2 )]′} tl21 l l rl21 ̃ E = B{ϵ1jl (ρ1)[ρ2 jl (ρ2 )]′ − ϵ2jl (ρ2 )[ρ1jl (ρ1)]′} ̃ E = Bϵ2ρ2 {hl(+)(ρ2 )j′(ρ2 ) − j (ρ2 )[hl(+)(ρ2 )]′} tl21 l l

n ,o l (+) Elm ̂ M(r) = i Lhl (k nr )Ylm(r), n ,o Elm E(r) = −

where

il+1 ∇ × Lhl(+)(knr )Ylm(r), ̂ k

A = [ρ1hl(+)(ρ2 )jl′(ρ1) − ρ2 jl (ρ1)[hl(+)(ρ2 )]′]−1 , B = [ϵ2hl(+)(ρ2 )[ρ1jl (ρ1)]′ − ϵ1jl (ρ1)[ρ2 hl(+)(ρ2 )]′]−1

and incident spherical waves,

ρ1 = (ωa /c) ϵ1 , ρ2 = (ωa/c) ϵ2 , and the prime denotes differentiation with respect to ρ1 and ρ2. The electric field within each layer n is then expanded as a sum over the above spherical waves. We use the convention E(r, t) = E(r)e−iωt + E*(r)eiωt, with the field amplitude written as

n ,i l Elm ̂ M(r) = i Ljl (k nr )Ylm(r), n ,i Elm E(r) = −

il+1 ∇ × Ljl (knr )Ylm(r)̂ k

which we label with orbital angular momentum numbers l = 1, 2, ... and m = −l, ···, l, as well as with an index ν = E and M for transverse electric and magnetic polarization, respectively. Here, L = −ir × ∇ is the angular momentum operator, h+l = −yl + ijl and jl are spherical Hankel and Bessel functions,84 Ylm are spherical harmonics, k = ω/c is the light wave vector in free space, kn = (ω /c) ϵn is the light wave vector in the medium, and ϵn is the permittivity. Scattering at a spherical interface (r = a) formed between media n (for r < a) and n + 1 (for r > a) can be easily described in terms of Mie scattering n,n+1 n+1,n n+1,n ̃ , implicitly defined through the coefficients rn,n+1 lν , tlν , r̃lν , and tlν expressions for the electric field produced upon incidence from either the inner or the outer side of the interface (see Figure 5a):

E(r) =

∑ [almnν Elmn,oν(r) + blmnν Elmn,iν(r)] lmν

We obtain the expansion coefficients anlmν and bnlmν by solving the linear set of equations n n,n+1 n n+1 blm almν + tlñν+ 1, nblm ν = rlν ν , n+1 n,n+1 n n+1 alm almν + rlñν+ 1, nblm ν = tlν ν

resulting from expressing the components leaving the n/(n + 1) interface in terms of the scattering of the waves coming into it. The source is provided by the external illumination in the surrounding 7920


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ACS Nano medium, for which we consider an incident plane wave, whose spherical-wave decomposition can be found elsewhere using the same notation as above.83 From the self-consistently obtained electric field we compute the absorbed power in layer n,

Pn =

ω Im{ϵn(ω)} 2π

lattice coupling to the dielectric lattice (ll). We denote the corresponding TBCs as Gelj,i/o and Gllj,i/o, respectively, where the i and o subscripts refer to the inner and outer metal/dielectric ll conductances for the lattice−lattice interfaces. The Gj,i/o channel are obtained by relying on the commonly used diffuse mismatch model, 61,90 which results in a temperatureindependent value Gllj,i/o = 141.5 MW m−2 K−1 (750 MW m−2 K−1) for interfaces formed between gold (aluminum) and silica.89 The lattice−lattice channel is usually regarded to be dominant over the electron−lattice one.91 Additionally, the TBCs are sensitive to the synthesis procedure,92,93 so that several previous works consider a temperature-independent total TBC as a simplifying assumption.90,92−95 Only a few experimental works isolate the effect of the electron−lattice channel,96−98 which combined with additional theoretical works89,99 show that it plays an important role in metal/ dielectric systems such as the ones here considered. We thus use an experimentally fitted function for the electron−lattice conductance of the interface between silica and gold:89,97

∫V d3r |E(r)|2 n

where the integral extends over the layer volume Vn. Noticing that the angular momentum operator L acts only on angular components of r, and using the orthogonality relations

∫ dΩr[LYlm(r)]̂ ·[LYl′m′(r)]̂ * = l(l + 1)δll′δmm′, ∫ dΩr[r ̂ × LYlm(r)]̂ ·[r ̂ × LYl′m′(r)]̂ * = l(l + 1)δll′δmm′, ∫ dΩr[LYlm(r)]̂ ·[r ̂ × LYl′m′(r)]̂ * = 0 the absorbed power can be calculated by performing the remaining radial integrals as ∞

Gjel,i/o = (96.12 + 0.189T je) MW m−2 K−1

(3) (see Figure 5c,d). Due to the scarcity of data for aluminum, we use the same expression for this material, which should provide a qualitative level of description. Finally, considering a total of N metal layers, we assume the outermost of them (j = N) to be in contact with the surrounding water environment. The corresponding TBC between the metal and water is assumed to have a temperature-independent value GllN,o = 105 MW m−2 K−1 (753 MW m−2 K−1) for gold57 (aluminum100). • Electron−lattice coupling inside the metal. We assume an electron−lattice coupling at the bulk of the metal proportional to the difference between electron and lattice temperatures, resulting in a power-density transfer gel(Tej − Tlj), with gel = 3 × 1016 W m−3 K−1 (3 × 1017 W m−3 K−1) for gold (aluminum).53 • Bulk thermal conductivity. We take the conductivities of silica and water as κsilica = 1.0 W m−1 K−1 and κwater = 0.6 W m−1 K−1. The metal conductivity is orders of magnitude larger (e.g., κgold = 318 W m−1 K−1), so we assume a uniform temperature inside each homogeneous metal region. With these parameters, we now write a self-consistent set of equations that express the condition of steady-state temperature distribution. First of all, the optical power Pj absorbed by electrons in the metal layer j (see eq 2),

l

ω Im{ϵn(ω)} ∑ ∑ l(l + 1) 2π l = 1 m =−l

Pn= ×

∫R

R n ,o n ,i

n n 2 (+) r 2dr[|alm Mhl (k nr ) + blm Mjl (k nr )|

n n 2 (+) ′ (knr ) + blm + |ϵn(ω)| |alm Ehl Ejl′(k nr )| ]

(2)

where Rn,i and Rn,o are the inner and outer radii of the layer under consideration, while the primes indicate derivation with respect to the argument. In our simulations, we take the permittivities from experimental data for gold,85 aluminum,86 and silica.87 We also assume ϵwater = 1.77 for the water environment. Incidentally, we have validated the results of this analytical method by comparing it to the boundary-element method,88 obtaining excellent agreement between both of them, although the present analytical approach is much faster. Heat Transfer Simulations. We evaluate the photothermal response of multishell nanostructures immersed in water similar to the one depicted in Figure 1a by adopting the two-temperature model and incorporating temperature-dependent TBCs. We focus on either gold or aluminum metal intercalated with silica layers. Under cw illumination conditions, light energy is absorbed by the electrons in the metallic shells and then partially transferred from those electrons to the atomic lattice (i.e., phonons) of both the metal and the adjacent dielectric layers. Additionally, phonons of the metal and dielectric regions exchange energy until a steady-state thermal distribution is established. The parameters that control these processes are illustrated in Figure 5b for a generic metallic shell j flanked by two dielectric layers. They are as follows: • Geometrical parameters. Each metal layer j has inner and outer radii Rj,i and Rj,o, while the corresponding metal/dielectric interfaces have areas Sj,i/o = 4πR2j,i/o and the metal layer volume is Vj = (4π/3)(R3j,o − R3j,i). • Temperatures. We consider the electron and lattice temperatures, Tej and Tlj, as well as the temperatures at the dielectrics right outside the inner and outer interfaces with the metal, Tdj,i and Tdj,o, respectively. The temperatures are assumed to be uniform within each metal layer because of its high thermal conductivity. In contrast, the temperature varies with radial distance in each dielectric layer (see below). • Thermal boundary conductances. When a temperature difference exists between both sides of an interface, heat flows through it with a power per unit area that is proportional to that difference. The coefficient of proportionality is the TBC, also known as Kapitza conductance.45−48 Two different channels of conductance are known to exist at metal/dielectric interfaces:89 metal electrons coupling to the dielectric lattice (el) and metal

Pj = g el Vj(T je − Tjl) + Gjel,iSj ,i(T je − T jd,i) + Gjel,oSj ,o(T je − T jd,o) (4) must be equal to the rate of heat transferred from the metal electrons to the lattices of both the metal (first term on the right-hand side of eq 4) and the adjacent dielectrics (rightmost two terms). The energy deposited by the electrons into the lattice of metal layer j, g el Vj(T je − Tjl) = Gjll,iSj ,i(Tjl − T jd,i) + Gjll,oSj ,o(Tjl − T jd,o)

(5)

is now exchanged with the adjacent dielectrics through the lattice− lattice TBC channel (right-hand side of eq 5). A similar balance applies to each dielectric layer, leading to

Sj ,o[Gjel,o(T je − T jd,o) + Gjll,o(Tjl − T jd,o)] = − Sj + 1,i[Gjel+ 1,i(T je+ 1 − T jd+ 1,i) + Gjll+ 1,i(Tjl+ 1 − T jd+ 1,i)]

(6)

where each side of the equation describes the power transferred from each of the two dielectric−metal interfaces. The condition of flux conservation across each dielectric layer leads to yet another set of equations: under stationary conditions the temperature obeys the Poisson equation,24 and thus, its radial dependence within a homogeneous dielectric layer has the form A + B/r, resulting in a diffused heat power 4πBκsilica; the coefficients A and B are directly related to the temperatures at the dielectric interfaces (i.e., at the radial 7921


Article

ACS Nano distances Rj,o and Rj+1,i for the dielectric layer sandwiched in between metal layers j and j + 1; see Figure 5b), whereas heat dissipation has to account for all of the light absorption in the metal layers surrounded j Pj′); we find by the dielectric shell under consideration (∑j′=1 j

∑ Pj ′ = j ′= 1

4πκsilicaR j + 1,iR j ,o R j + 1,i − R j ,o

aluminum instead of gold, and a comparison of two- and one-temperature models for the thermal response of gold (PDF)

AUTHOR INFORMATION

(T jd,o − T jd+ 1,i)

Corresponding Author

(7)

*E-mail: [email protected].

In this paper, we consider multishells formed by N = 1−3 metal layers, with a dielectric core and with an outermost metal layer directly in contact with the surrounding water medium. We then have to find the 4N temperatures Tej , Tlj, Tdj,i, and Tdj,o associated with each metal layer j = 1, ..., N and its surrounding dielectrics. These are the unknowns in the above equations. The number of equations are N in each set of eqs 4 and 5 (one per metal layer) and N − 1 in each set of eqs 6 and 7 (one per dielectric layer flanked by two metal layers), resulting in a total of 4N − 2 equations. The remaining two equations are provided by (i) heat balance of the entire particle, which determines the water temperature TdN,o right at the interface with the multishell according to24

ORCID

F. Javier García de Abajo: 0000-0002-4970-4565 Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work has been supported in part by the Spanish MINECO (MAT2014-59096-P and SEV2015-0522), AGAUR (2014 SGR 1400), Fundació Privada Cellex, and the National Natural Science Foundation of China (61425023). L.M. acknowledges support from China Scholarship Council.

N

∑ Pj = 4πκ waterRN ,o(TNd ,o − T0) j=1

REFERENCES

(8)

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where T0 = 300 K is the temperature of the environment and (ii) the specialization of eq 6 to the dielectric core, el d ll d G1,i (T1e − T1,i ) + G1,i (T1l − T1,i )=0

(9)

Equations 4−9 thus form a nonlinear set of equations (notice the temperature dependence of Gelj,i/o through eq 3) that we solve using the Newton−Raphson iteration method. Calculation of the Pressure at the Nano-Oven Core. Each curved interface introduces a pressure difference due to the surface tension. For solid−liquid and liquid−liquid interfaces, this is the socalled Laplace pressure, the increase of which at the interface between media 1 and 2 is given by 2γ12/R12, where R12 is the radius of curvature of the interface and γ12 is the interfacial energy. Instead, we are dealing with solid−solid interfaces, for which the expression above needs to be changed to Δpintrinsic = 2γ12/R12 + C, where both C and γ12 can be expressed in terms of the shear modulus of the outer medium G, the bulk modulus of the inner medium K, and the stress of the interface h, as discussed in the literature.101,102 More precisely, γ12 = mh, where m = 1/(1 + 4G/3K), while C = 4mεG originates in the deformation of the interface, with a characteristic strain that depends on actual size102 (we take ε ≈ −0.0017 as an average value that should be appropriate for the nanometer-sized layers). Plugging parameters for gold103 (Ggold = 27 GPa, Kgold = 180 GPa) and silica104 (Gsilica = 31 GPa, Ksilica = 37 GPa) and using a typical stress h ≈ 1 N/m,102 we find γ12 = 0.8 N/m and C = −0.2 GPa for a gold/silica (inner/outer) interface and γ12 = 0.5 N/m and C = −0.1 GPa for a silica/gold one. Both the constant and the 1/R12 terms of this intrinsic pressure contribute with values on the order of the GPa. Another major source of pressure originates in thermal stress, which has been recently measured in core−shell nanoparticles.105 Because the linear thermal expansion coefficient β is about 25 times larger in gold than in silica, we can neglect the effect of the dielectric and estimate this source of pressure increase as Δpthermal = 3βgold(Tgold − T0) Kgold using βAu = 14.2 × 10−6 K−1105 (e.g., we have Δpthermal ≈ 4 GPa for Tgold − T0 = 500 K). The expression Δpintrinsic + Δpthermal is used in this work to estimate the concatenated increases of pressure in Figure 1. Incidentally, Δpthermal is dismissed for the outermost layer because it can freely expand.

ASSOCIATED CONTENT S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.7b02426. Additional details on the geometrical optimization of the nano-oven, calculations similar to those of Figure 3 using 7922


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